Homotopy sphere

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Short description: Concept in algebraic topology: n-manifold with the same homotopy and homology groups as the n-sphere

In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homology sphere.[1]

The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to the n-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005.

The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that homotopy spheres in those dimensions are precisely exotic spheres. It is still an open question ((As of February 2019)) whether or not there are non-trivial smooth homotopy spheres in dimension 4.

References

  1. A., Kosinski, Antoni (1993). Differential manifolds. Academic Press. ISBN 0-12-421850-4. OCLC 875287946. http://worldcat.org/oclc/875287946. 

See also